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- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/09%3A_Linear_Higher_Order_Differential_Equations/9.01%3A_Introduction_to_Linear_Higher_Order_EquationsThis section presents a theoretical introduction to linear higher order equations. We will sketch the general theory of linear n-th order equations.
- https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/09%3A_Linear_Systems_of_Differential_Equations/9.03%3A_Basic_Theory_of_Homogeneous_Linear_SystemsIn this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with...In this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with the theory of linear homogeneous scalar equations.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/09%3A_Linear_Higher_Order_Differential_Equations/9.01%3A_Introduction_to_Linear_Higher_Order_EquationsThis section presents a theoretical introduction to linear higher order equations. We will sketch the general theory of linear n-th order equations.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/08%3A_Back_to_Power_Series/8.04%3A_Boundary_Issues_and_Abel%E2%80%99s_TheoremThe integrations we performed in Chapter 2 are legitimate due to the Abel's theorem which extends uniform convergence to the endpoints of the interval of convergence even if the convergence at an endp...The integrations we performed in Chapter 2 are legitimate due to the Abel's theorem which extends uniform convergence to the endpoints of the interval of convergence even if the convergence at an endpoint is only conditional. Abel did not use the term uniform convergence, as it hadn’t been defined yet, but the ideas involved are his.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/10%3A_Linear_Systems_of_Differential_Equations/10.03%3A_Basic_Theory_of_Homogeneous_Linear_SystemsIn this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with...In this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with the theory of linear homogeneous scalar equations.
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/04%3A_Linear_Second_Order_Equations/4.02%3A_Introduction_to_Linear_Higher_Order_EquationsThis section presents a theoretical introduction to linear higher order equations. We will sketch the general theory of linear n-th order equations.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/19%3A_Linear_Higher_Order_Differential_Equations/19.01%3A_Introduction_to_Linear_Higher_Order_EquationsThis section presents a theoretical introduction to linear higher order equations. We will sketch the general theory of linear n-th order equations.
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/10%3A_Linear_Systems_of_Differential_Equations/10.03%3A_Basic_Theory_of_Homogeneous_Linear_SystemsIn this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with...In this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with the theory of linear homogeneous scalar equations.
- https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/3%3A_Second_Order_Linear_Differential_Equations/3.6%3A_Linear_Independence_and_the_WronskianThis is a system of two equations with two unknowns. The determinant of the corresponding matrix is the Wronskian. Hence, if the Wronskian is nonzero at some t0 , only the trivial solution exists. Hen...This is a system of two equations with two unknowns. The determinant of the corresponding matrix is the Wronskian. Hence, if the Wronskian is nonzero at some t0 , only the trivial solution exists. Hence they are linearly independent.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/07%3A_Linear_Higher_Order_Differential_Equations/7.01%3A_Introduction_to_Linear_Higher_Order_EquationsIt’s easy to show that if y1, y2, …, yn are solutions of the homogeneous equation on (a,b), then so is any linear combination of {y1,y2,…,yn}. (See the proof of Theorem...It’s easy to show that if y1, y2, …, yn are solutions of the homogeneous equation on (a,b), then so is any linear combination of {y1,y2,…,yn}. (See the proof of Theorem 5.1.2.) We say that {y1,y2,…,yn} is a fundamental set of solutions of the homogenous equation on (a,b) if every solution on (a,b) can be written as a linear combination of n linearly independent solutions {y1,y2,…,yn}, as in Equation ???. In this ca…
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/10%3A_Linear_Systems_of_Differential_Equations/10.03%3A_Basic_Theory_of_Homogeneous_Linear_SystemsIn this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with...In this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with the theory of linear homogeneous scalar equations.
